title: Trading spaces: building three-dimensional nets from two-dimensional tilings.
abstract: We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S (2)), Euclidean (E (2)) and hyperbolic (H (2)) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.
authors.affiliation: Department of Applied Mathematics, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Captial Territory 0200, Australia.
authors.forename: Toen
authors.initials: T
authors.lastname: Castle
authors.forename: Myfanwy E
authors.initials: ME
authors.lastname: Evans
authors.forename: Stephen T
authors.initials: ST
authors.lastname: Hyde
authors.forename: Stuart
authors.initials: S
authors.lastname: Ramsden
authors.forename: Vanessa
authors.initials: V
authors.lastname: Robins
keywords: symmetry; three-dimensional tiling; two-dimensional nets